Finding flavons at colliders

We conduct a comprehensive investigation into the flavour phenomenology and collider signatures of flavon of $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetries for the soft symmetry-breaking scenario and a new symmetry-conserving mechanism at the high-luminosity LHC, high energy LHC, and a 100 TeV hadron collider. The flavour physics of quark and leptonic observables places different bounds on the parameter space of flavons of $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetries. On the collider side, the decay $t \rightarrow c a$ can be probed by the high-luminosity LHC, high energy LHC, and a 100 TeV hadron collider for the $\mathcal{Z}_{\rm 8} \times \mathcal{Z}_{\rm 22}$ flavour symmetry. The inclusive production signatures can be used to probe the flavon of all the $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries for the soft symmetry-breaking scenario for a heavy flavon at a 100 TeV collider. Flavons of all the $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries can be probed at high energy LHC and a 100 TeV collider for a low mass in the case of soft symmetry-breaking. The di-flavon production is within reach of the high-luminosity LHC, high energy LHC, and a 100 TeV collider only for a light flavon. The 14 TeV high-luminosity LHC can probe only the $\mathcal{Z}_{\rm 2} \times \mathcal{Z}_{\rm 5}$ and $\mathcal{Z}_{\rm 8} \times \mathcal{Z}_{\rm 22}$ flavour symmetries for a few specific inclusive signatures. The symmetry-conserving scenario remains beyond the detection capabilities of any collider.